$ E = \left[\begin{array}{rr}1 & 4 \\ 1 & -1\end{array}\right]$ $ B = \left[\begin{array}{rrr}-1 & 5 & 3 \\ 2 & -1 & 0\end{array}\right]$ What is $ E B$ ?
Answer: Because $ E$ has dimensions $(2\times2)$ and $ B$ has dimensions $(2\times3)$ , the answer matrix will have dimensions $(2\times3)$ $ E B = \left[\begin{array}{rr}{1} & {4} \\ {1} & {-1}\end{array}\right] \left[\begin{array}{rrr}{-1} & \color{#DF0030}{5} & \color{#9D38BD}{3} \\ {2} & \color{#DF0030}{-1} & \color{#9D38BD}{0}\end{array}\right] = \left[\begin{array}{rrr}? & ? & ? \\ ? & ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ E$ , with the corresponding elements in column $j$ of the second matrix, $ B$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ E$ with the first element in ${\text{column }1}$ of $ B$ , then multiply the second element in ${\text{row }1}$ of $ E$ with the second element in ${\text{column }1}$ of $ B$ , and so on. Add the products together. $ \left[\begin{array}{rrr}{1}\cdot{-1}+{4}\cdot{2} & ? & ? \\ ? & ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ E$ with the corresponding elements in ${\text{column }1}$ of $ B$ and add the products together. $ \left[\begin{array}{rrr}{1}\cdot{-1}+{4}\cdot{2} & ? & ? \\ {1}\cdot{-1}+{-1}\cdot{2} & ? & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ E$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ B$ and add the products together. $ \left[\begin{array}{rrr}{1}\cdot{-1}+{4}\cdot{2} & {1}\cdot\color{#DF0030}{5}+{4}\cdot\color{#DF0030}{-1} & ? \\ {1}\cdot{-1}+{-1}\cdot{2} & ? & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rrr}{1}\cdot{-1}+{4}\cdot{2} & {1}\cdot\color{#DF0030}{5}+{4}\cdot\color{#DF0030}{-1} & {1}\cdot\color{#9D38BD}{3}+{4}\cdot\color{#9D38BD}{0} \\ {1}\cdot{-1}+{-1}\cdot{2} & {1}\cdot\color{#DF0030}{5}+{-1}\cdot\color{#DF0030}{-1} & {1}\cdot\color{#9D38BD}{3}+{-1}\cdot\color{#9D38BD}{0}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rrr}7 & 1 & 3 \\ -3 & 6 & 3\end{array}\right] $